PhD Oral Exam - Oliver Russell, Mathematics

When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.

Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.

Abstract

The long-standing Gaussian product inequality (GPI) conjecture states that E[|X1|y1|X2|y2···|Xn|yn] ≥ E[|X1|y1]E[|X2|y2]···E[|Xn|yn] for any centered Gaussian random vector (X1 , . . . , Xn) and any non-negative real numbers yj , j = 1, . . . , n. First, we complete the picture of bivariate Gaussian product relations by proving a novel “opposite GPI” when -1<y1<0 and y2>0: E[|X1|y1|X2|y2] ≤ E[|X1|y1]E[|X2|y2]. Next, we investigate the three-dimensional inequality E[X12X22m2Xn2m3] ≥ E[X12]E[X22m2]E[Xn2m3] for any natural numbers m2, m3. We show that this inequality is implied by a combinatorial inequality which we verify directly for small values of m2 and arbitrary m3. Then, we complete the proof through the discovery of a novel moment ratio inequality which implies this three-dimensional GPI. We then extend these three-dimensional results to the case where the exponents in the GPI can be real numbers rather than simply even integers. Finally, we describe two computational algorithms involving sums-of-squares representations of polynomials that can be used to resolve the GPI conjecture. To exhibit the power of these novel methods, we apply them to prove new four- and five-dimensional GPIs.